For compound (A)n(B)m we
can expect ionic bonding to predominate when atom A has low electronegativity
and atom B has a high electronegativity. In this case electron transfer
from one atom to another leads to the formation of A+B-.
For the main group elements the electron transfer continues until the ions
have closed shell configurations.

For ionic compounds the bonding forces are electrostatic
and therefore omni-directional. The bonding forces should be maximized
by packing as many cations around each anion, and as many cations around
each anion as is possible. The number of nearest neighbor ions of
opposite charge is called the coordination number. We must
realize however that the coordination numbers are constrained by the stoichiometry
of the compound and by the sizes of the atoms.

e.g. For sodium chloride, Na+Cl-,
there are 6 anions around each cation (coordination number Na = 6); because
of the 1:1 stoichiometry there must also be 6 Na cations around each Cl
anion. For Zr4+O2-2 there are 8
anions around each cation, therefore there must be only 4 cations around
each anion.

Simple ionic crystal structures can be approached
in terms of the close packing procedures developed for metallic structures.
In most (but by no means all) ionic compounds the anions are larger than
the cations. In these cases it is possible to visualize the structures
in terms of a close packed arrangement of the larger anions, with the cations
occupying the vacant interstices between the close packed layers.
Recall that although ccp & hcp are the most efficient ways of
packing spheres, only 74% of the available space is filled, the 26% "free
space" is in the form of different types of holes or sites which can be
occupied by the smaller cations in the ionic structures .

First let us consider the types of holes available
in a close packed anion arrangement.

(return to top)Types
of cations sites available in close packed anion arrays.

As shown below, the stacking of two close packed
anion layers produces 2 types of voids or holes. One set of holes
are octahedrally coordinated by 6 anions, the second set are tetrahedrally
coordinated by 4 anions. One octahedral site and two tetrahedral
sites are created by each anion in the close packed layer.

Having determined what types of holes are available
we must now decide:

(a) Which sites are occupied by a given cation.
This determined by the radius ratio (= rcation/ranion)(b) How many sites are occupied. This is
determined by the stoichiometry.

Which sites ?: Radius Ratio rules.

The relative sizes of the anions and cations required
for a perfect fit of the cation into the octahedral sites in a close packed
anion array can be determined by simple geometry:

Similarly for a perfect fit of a cation into the
tetrahedral sites it can be shown that rcation/ranion
= 0.225.

For these two "ideal fit" radius ratios the anions
remain close-packed.(return to top)

Stable Bonding Configurations in Ionic solids.

In reality an ideal fit of a cation into the close
packed anion arrangement almost never occurs . Now consider
what would be the consequence of placing a cation that is (a) larger than
the ideal, (b) smaller than the ideal, into the cation sites.

For a stable coordination the bonded cation and anion must be in
contact with each other.

If the cation is larger than the ideal radius ratio value the cation
and anion remain in contact, however the cation forces the anions apart.
This is not a problem as there is no need for the anions to remain in contact.

If the cation is too small for the site then the cation would "rattle"
and would not be in contact with the surrounding anions. This is
an unstable bonding configuration.

Note however in a few rare cases solids do contain cations that are
too small for their sites, in these cases the cation moves off the center
of the site and adopt a distorted octahedral coordination. These
solids typically exhibit novel properties, such as, for example,
ferroelectricity and piezoelectricity.
(return to top)

Summary, Radius ratio
rules for close packed anion structures.

Ccp Anion Packing: Examples.

Now we know how to determine which sites will be filled, we place the
appropriate number of cations into the structure, making sure that we observe
the correct stoichiometry. The figure below shows a view of the octahedral
and tetrahedral interstices that are available in the fcc cell of a ccp
anion arrangement . By filling these to differing degrees a
number of very common types of crystal structures can be produced.

The figure below shows the filling of the octahedral sites by Na
(green) within the ccp (ABCABC) anion (red) array.

This figure shows the face centered cubic unit cell of the NaCl
structure.

Note that there is an fcc arrangement of the Na cations and Cl anions.
We can make sense of the cell contents using our knowledge of lattices.
For an fcc lattice there are 4 lattice points per cell, the motif in this
case is a Na cation and a Cl anion. Therefore the cell contents are
4 Na cations + 4 Cl anions.

Which sites are filled ?: see picture below.
Note the filling of diagonally opposite sites to maximize the cation-cation
separations .

Again the lattice is fcc, the motif consists of 1 S and 1 Zn.
(return to top)

HCP ANION PACKING

It is also possible to follow the simple methods used above to fill
the octahedral and tetrahedral sites in an hcp (ABAB) array of anions.
Again many common structure types can be generated. These will not
be considered in this class.
(return to top)

CUBIC ANION PACKING

In these structures the anions are not close packed, but occupy
just the corners of a cube. In this case the center of the cube (surrounded
by 8 anions) can be occupied by a suitably sized cation. This site
is larger than the tetrahedral or octahedral positions in the close packed
structures. The radius ratio for a perfect fit of a cation in a cubic
site can again be calculated using simple geometry and is 0.73. This
structure should therefore be adopted when the rcation/ranion
is equal to or greater than 0.73.

By including the possibility of cubic coordination we can now complete
our table for predicting cation coordinations from the radius ratio rules:

One cubic site per F anion; from stoichiometry only 50% cubic sites
filled by Ca cations.
Arrangement of the filled cubic sites is such that the Ca-Ca distances
are as large as possible (compare the Ca distribution to that of Zn in
ZnS)
Coordination numbers: Ca2+ surrounded by 8 F- 's;
F- surrounded by 4 Ca2+'s.
Other examples: ZrO2,